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A quantum secure direct communication protocol based on four-qubit cluster state

Authors


ABSTRACT

We propose a quantum secure direct communication protocol utilizing four-qubit cluster state to enhance the efficiency of eavesdropping detection. In the security analysis, by applying the method of the entropy theory, we contrast our scheme to another two strategies, the Ping-pong protocol and the protocol using two particles of Einstein–Podolsky–Rosen pair as detection particles. The comparison results show that if the eavesdropper obtains the same amount of information, the presented quantum protocol strategy will have a larger detection probability than the other two. At last, the security of the proposed protocol is discussed. The analysis results indicate that the protocol in this paper is more secure. Copyright © 2013 John Wiley & Sons, Ltd.

1 INTRODUCTION

Quantum secure communication consists of quantum key distribution (QKD) and quantum secure direct communication (QSDC). This paper mainly discusses the quantum secure direct communication. Quantum mechanics offers some unique capabilities for processing and transmitting quantum information. Over the past decade, scientists have made dramatic progress in the field of quantum communication. Since Bennett and Brassard [1] proposed the pioneer QKD protocol in 1984 in which two remote authorized users (Alice and Bob) can create a shared private key, many quantum information security processing schemes have been presented, such as quantum teleportation [2-4], quantum dense coding [5, 6], DSQC [7, 8], and QSDC [9-13].

In 2002, Boström and Felbinger proposed a Ping-pong protocol using Einstein–Podolsky–Rosen (EPR) pairs as quantum information carriers [9]. But it proved to be an only deterministic QKD scheme rather than a QSDC scheme. Later, researchers are interested in QSDC, and many protocols were proposed, including the protocols without using entanglement [14, 15], the protocols using entanglement [16-21], and the two-way QSDC protocols [22-31]. In these protocols, the secret message is transmitted through the transmission channel directly. Compared with the QKD, the security requirement of QSDC is higher, since the secret message cannot be leaked out absolutely in QSDC. For example, there will be some security problems when the “Ping-pong” protocol is used for QSDC [32-35]. But the unconditional security can still be achieved by adopting a well-designed QSDC protocol in theory [36-38].

Long et al. proposed a theoretical two-step QKD scheme using EPR pairs [10], which is the first QSDC protocol. It introduced the method of quantum data block transmission for the security in QSDC based on error rate analysis. To guard the secret message, one has to ensure the security of a block of quantum data [8-13] before encoding the secret message. When errors exist, error correction and quantum privacy amplification can be used to maintain the security of the QSDC protocol. In 2003, based on the basic idea in Ref. [10], Deng et al. proposed a two-step secure QSDC scheme with EPR pairs [11] transmitted in blocks as well. Moreover, error correction and quantum privacy amplification can be used to maintain the security of the QSDC scheme.

This paper presents an improved protocol based on four-qubit cluster state to increase the efficiency of eavesdropping detection in the QSDC protocol. We call our proposed scheme in this paper FPP for short, while the original Ping-Pong protocol In Ref [13] is called OPP, and the protocol proposed in Ref [13], used two particles of EPR pair as detection particles is called MPP by its author. We use the method of entropy theory to compare different detection strategies, and three detection strategies are compared. The comparison results show that the detection rates of OPP, MPP and FPP are 50%, 50% and 75% respectively, when the eavesdroppers get full information. In the end, the security of the proposed protocol is discussed. The analysis results show that the proposed protocol is more secure than the other two.

2 RELATED WORKS

2.1 The original “Ping-pong” protocol

The basic principle of the OPP, which is presented by Boström and Felbinger in 2002 [9], is that 1-bit information can be encoded in the states |ψ±〉, which is completely unavailable to anyone who has access to one qubit in the state |ψ+〉 or |ψ〉. To get information from Alice in secret, Bob prepares two photons in the Bell state |ψ+〉 at first. He stores one photon, the home qubit, and sends the other, the travel qubit, to Alice through a quantum channel. In message mode, on receiving the travel qubit, Alice takes a coding operation either I or σz on the travel qubit, then sends it back to Bob. Bob, who has now both qubits again, performs a Bell measurement on both qubits, resulting in the final state |ψ+〉 or |ψ〉, and he decodes the message as |ψ+〉 ⇒ 0, |ψ〉 ⇒ 1; thus, he has received one bit of information from Alice. To detect eavesdropping, Alice can switch to control mode with probability c or message mode with probability 1 − c in random. In control mode, Alice measures the polarization of the travel qubit in the Z-basis Bz = {|0〉, |1〉} and broadcasts the result using the public channel. Receiving the result from Alice, Bob also switches to control mode and measures the home qubit in the same basis Bz and compares both measuring results, which should be opposite in the absence of Eve. If both measuring results coincide, Bob knows that Eve is in the line and stops the communication.

2.2 The MPP protocol in [13]

The basic principle of the eavesdropping detection strategy in the MPP protocol in [13] is that in control mode, Bob sends two qubits in the Bell state |ψ+〉 to Alice. After Alice receives both particles, she performs a Bell measurement. If there is no eavesdropper, each measuring result should be in the Bell state |ψ+〉. Otherwise, the communication is interrupted.

Suppose that the message transmitted is in a sequence xN = (x1, …,xN), where xi ∈ {0,1}, i = 1, 2, …, N. Now, let us give an explicit introduction of MPP [13].

  • (S1) Bob prepares a number of quantum bits as follows.
    • (i) Bob prepares a large number (N) of Bell states |ψ+〉 in sequence. He extracts all the first particles in the Bell states, forming a series of particles S1 (the travel qubits) in order, S1 is used to transmit message, and the remaining particles in the Bell states form a series of particles S2 (the home qubits). This step corresponds to message mode in the OPP.
    • (ii) Bob prepares a large number (cN/(1 − c)) of Bell states |ψ+〉, forming a series of particles S3 in order, S3 is used to detect eavesdropping, and this step corresponds to control mode in the OPP. Here, c expresses the probability of entering into control mode in OPP [9]. Note that S3 includes 2cN/(1 − c) quantum bits.
    • (iii) Bob inserts decoy photon S3 to particles S1 in random. Then, a new sequence S4 containing Bell type decoy photons is produced, Note that only Bob knows decoy photons' positions.
  • (S2) Bob stores particles S2 and sends particles S4 to Alice.
  • (S3) After Alice receives particles S4, Bob tells her the positions where the decoy photons are.
  • (S4) Alice extracts all the decoy photons from particles S4, forming a series of qubits S5, and performs a Bell state measurement. If there is no eavesdropper, every decoy photons should be in the Bell state |ψ+〉. Otherwise, the communication stops.
  • (S5) Define C0 = I and C1 = σz. Alice performs the coding operation C0 or C1 on S5 with probability p0 and p1, respectively, and then sends S5 back to Bob.
  • (S6) Receiving the encoded particles, Bob performs Bell state measurements on particles from S2 and S5. Every measurement result should be either |ψ+〉 or |ψ〉, which encodes 0 and 1, respectively. Now, the communication ends successfully.

3 THE FPP PROTOCOL

3.1 The process of FPP protocol

In the protocol in [10], the transmission is managed in batches of N EPR pairs. An advantage of batch transmission scheme is that we can check the security of the transmission by measuring some of the decoy photons in the first communication step, where both Alice and Bob contain a particle sequence at hand, In the first communication step, an eavesdropper has no access to the first particle sequence, and then no information will be leaked to Eve, whatever Eve has done to the second particle sequence. Following this method using block transmission, the FPP scheme is proposed.

Suppose that the message transmitted is in a sequence xN = (x1, …,xN), where xi ∈ {0,1}, i = 1, 2, …, N.

Define

display math(1)
display math(2)

Now, let us give all the details of FPP scheme.

  • (S1) Bob prepares Bell states and inserts enough four-qubit cluster states as follows.
    • (i) Bob prepares a large number (N) of Bell states |Φ+〉 in sequence. He extracts all the first particles in the Bell states, forming a series of particles A (the travel qubits) in order, and A is used to transmit secure message. The remaining particles in the Bell states form a series of particles B (the home qubits). Thus, this step corresponds to the message mode in OPP.
    • (ii) Bob prepares a large number (cN/(1 − c)) of ordered four-qubit cluster states |ψ〉 and forms a series of particles C to detect eavesdropping, corresponding to control mode in OPP. Note that particles C includes 4cN/(1 − c) qubits. Bob inserts particles of four-qubit cluster state to particles A randomly, forming a new sequence D, which contains decoy photons of four-qubit cluster state. Note that only Bob knows the positions of decoy photons. Bob stores particles B and sends particles D to Alice.
  • (S2) The eavesdropping detection by Alice.
    • After Alice receives particles D, Bob tells her the positions of decoy photons of four-qubit cluster state in C. Alice extracts the decoy photons from particles D and performs four-particle cluster state measurement. If there is no eavesdropper, each result should be in the four-particle cluster state, and the FPP protocol continues. Otherwise, the communication is interrupted, and the FPP protocol switches to (S1).
  • (S3) Alice encodes her ciphertext and inserts decoy photons of four-qubit cluster states randomly like Bob.
    • Define C0 = I and C1 = σz. Alice performs the coding operation C0 or C1 on the remaining particles of D with probabilities p0 and p1, respectively, Alice inserts four-qubit cluster state into particles D randomly, forming a new sequence E. E contains decoy photons of four-qubit cluster state, but only Alice knows the positions of decoy photons. Alice sends E back to Bob.
  • (S4) The eavesdropping detection by Bob.
    • Similar to step S2, on receiving particles E, Bob extracts the decoy photons from particles E and performs four-qubit cluster state measurement to ensure the security of the quantum channel. If there is no eavesdropper, Bob starts to decode the ciphertext and the remaining particles names F. Otherwise, the communication is interrupted, and the FPP protocol stopped.
  • (S5) Bob decodes the ciphertext.
    • Bob measures the state of particles B and F in the Bell basis state. There are only two possible outcomes of the measurement, |ψ+〉 or |ψ〉, which encodes 0 or 1, respectively. The FPP protocol ends successfully.

3.2 The security analysis of FPP protocol

In the OPP [9], the author Boström calculates the maximal amount of the information I(dlO) that Eve can eavesdrop and the probability dlO that Eve is detected. And the function I(dlO) is provided as follows.

When p0 = p1 = 1/2,

display math(3)

The above method can be used to compare the efficiency of different protocols used in eavesdropping detection.

In the MPP, the maximal amount of the information I(dlM) that Eve can eavesdrop is

display math(4)

where

display math(5)

and dlM is the probability of Eve being detected.

Now, let us analyze the efficiency of the eavesdropping detection in FPP protocol. In order to gain the information Alice operates on the travel qubits, Eve performs a unitary attack operation math formula on the communication system at first. Then, Alice takes a coding operation on the travel qubits. Eve performs a measurement on the composed system at last. Note that, in our proposed protocol, all the transmitted particles are sent in block before detecting eavesdropping, which is different from OPP. Since Eve does not know which particles are used to detect eavesdropping, what she can do is only performing the same attack operation on all the particles. As for Eve, the state of the travel qubits is indistinguishable from the communication system, then all the travel qubits are considered in either of the states |0〉 or |1〉 with equal probability p = 0.5.

Generally speaking, suppose that there is a group of decoy photons the state of four-qubit cluster states |ψ〉, and after the attack operation math formula, the states |0〉 and |1〉 become

display math(6)
display math(7)

where |xi〉 and |yi〉 are the pure ancillary states determined by math formula uniquely and

display math(8)

Let us calculate the detection probability. Attacked by Eve, the state of the communication system becomes

display math(9)

Obviously, when the measurement is taken on the decoy photons, the probability without eavesdroppers is

display math(10)

So, the lower bound of the detection probability(dlF)is

display math(11)

Now, let us analyze how much information Eve can gain maximally when there is no control mode run. Suppose that |α|2 = a, |β|2 = b, |m|2 = s, |n|2 = t, where a, b, s, and t are positive real numbers, and a + b = s + t = 1. Then,

display math(12)

In the case of p0 = p1 = 0.5, when Bob sends |0〉 to Alice, the maximal amount of information is equal to the Shannon entropy of a binary channel,

display math(13)

Assume that Bob sends |1〉 rather than |0〉. The above security analysis can be done in full analogy, resulting in the same crucial relations. The maximal amount of information is equal to the Shannon entropy of a binary channel,

display math(14)

So, the maximal amount of information Eve can obtain is

display math(15)

After some simple mathematical calculations, when a = t, we can get

display math(16)

and the maximum I is

display math(17)

The aforementioned analysis shows that function I(dlO), I(dlM), and I(dlF) have the similar algebraic properties. If Eve gains full information (I = 1), the probabilities of eavesdropping detection are dlO (I = 1) = 0.5 and dlM (I = 1) = 0.5 in the OPP and the MPP separately, while in the FPP, dlF (I = 1) = 0.75.

We contrast three functions in, Figure 1 and Table 1. As shown in Figure 1 and Table 1, if Eve gains the same amount of information, she must face a larger detection probability in the FPP than in other two protocols. The comparison results also indicate that the FPP is more secure than the other two. Of course, in order to detect eavesdropping, Bob needs send 4cN/(1 − c) particles, while only send 2cN/(1 − c) particles in MPP, and only send cN/(1 − c) particles in OPP. In other words, Bob gains the better security at the cost of sending more particles.

Figure 1.

The comparison of the three detection results.

Table 1. The accurate comparison figures.
dProtocol (I/bit)
OPPMPPFPP
0.20.030.060.12
0.40.080.150.27
0.60.150.250.44
0.80.240.370.60
1.00.500.500.75

In Figure 1, the dotted line expresses the function I(dlO) in the OPP, the thin line expresses the function I(dlM) in the MPP, and the thick line expresses the function I(dlF) in the FPP. Obviously, if Eve wants to have the same amount information, she must encounter higher detection efficiency in FPP. Also, if there is the same detection efficiency, Eve will eavesdrop less information.

When we take into account probability c in control mode, we found that the effective transmission rate, that is, the number of message bits per protocol transmit, is 1 − c, equal to the probability for a message transfer. Thus, if Eve wants to eavesdrop one transferred message without being detected, the probability for this event reads is as follows

display math(18)

Thus, the probability of successfully eavesdropping eavesdrop I = nI(d) bits reading s(I,c,d) = s(c,d)I/I(d) as follows

display math(19)

where

display math(20)

Note that when I →  (a message or key of infinite length), we have s → 0, Therefore, the presented protocol in this paper is asymptotically secure. If the security of the quantum channel is ensured, the protocol is completely secure. For example, In Figure 2, we plot the successful eavesdropping probability, which is a function of the information gain I, and the control mode is c = 0.5. Note that for d < 0.5, Eve only gets part of the message, and he even could not know which part the acquired message is.

Figure 2.

Eavesdropping success probability.

4 CONCLUSIONS

This paper introduces a QSDC protocol using four-qubit cluster state, and compares three eavesdropping detection strategies. The analysis shows that if the eavesdropper obtains the same amount of information, she must face a larger detection probability in the FPP than the other two, the comparison results show that the detection efficiency in FPP is higher than that in OPP and MPP. Therefore, FPP can ensure the “Ping-pong” protocol more secure. We also found that in order to detect eavesdropping, Bob needs send 4cN/(1 − c) particles, while the OPP needs send no extra particles and the MPP needs send 2cN/(1 − c) particles. In other words, Bob gains better security at the cost of sending more particles. However, suppose that there are 400 decoy photons used in eavesdropper detection, the proposed protocol only needs to send 100 four-qubit cluster states, where the MPP (uses EPR pair as detection particles) has to send 200 EPR pairs. In further work, the security improvement of QSDC protocol will be researched.

ACKNOWLEDGEMENT

This work is supported by the National Natural Science Foundation of China (grant nos. 61100205 and 61100208).

Ancillary